Fragment Localized Molecular Orbitals

We introduce the concept of fragment localized molecular orbitals (FLMOs), which are Hartree–Fock molecular orbitals localized in specific fragments constituting a molecular system. In physical terms, we minimize the local electronic energies of the different fragments, at the cost of maximizing the repulsion between them. To showcase the approach, we rationalize the main interactions occurring in large biological systems in terms of interactions between the fragments of the system. In particular, we study an anticancer drug intercalated within DNA and retinal in anabaena sensory rhodopsin as prototypes of molecular systems embedded in biological matrixes. Finally, the FLMOs are exploited to rationalize the formation of two oligomers, prototypes of amyloid diseases, such as Parkinson and Alzheimer.

S1 Computational Protocol S1.1 Non-Covalently Bonded Fragments BC A A B C 1 STEP 2 STEP Figure S1: Three fragment scheme Let us consider the three fragment scheme in Fig. S1. Let us divide the system in the parts: A and B + C. Then, the A density can be localized by following MLHF-AB scheme. In particular, the sum of the energies of the two parts reads: where, BC = B + C, and the same holds for the corresponding density matrices. By using D = D A + D BC then, eq. S1 becomes: which is exactly the same formulation proposed for MLHF-AB. After minimizing eq. S3,

S2
we obtain two densities for A and BC systems which are maximally localized. However, B and C are not maximally localized in their specific spatial regions. To this end, we can now reformulate the problem as a true three fragment scheme, in which, however, D A is kept frozen, and its orbitals do no enter in the occupied occupied rotations, which will be performed in the BC space only. However, the effect of D A is taken into account by considering Eq. S4, which can be recasted in the following way: from which the Fock matrix can be constructed:  Figure S2: Example of electron assignment in Hexane as partitioned in three fragments.
We now consider the case in which the system is a supramolecule in which the different fragments are covalently bonded, via a single covalent bond. As explained in the main text, our procedure is carried out for closed shell fragments. Therefore, the user needs to define S3 to which fragment the bonding electrons have to be assigned. As an example, we show how such a partitioning can be done for the Hexane molecule in Fig. S2. In this case, we assign the bonding electrons between frag1 and frag2 to frag1, and those between frag2 and frag3 to frag2. Notice that the procedure is completely independent of such an assignment (vide infra). Once such a decision is taken, the computational protocol exactly follows what has already been explained in case of the water trimer. At the end of the procedure, there will be three fragment densities characterized by the pre-defined number of electrons. More importantly, for the fragments to which the bonding electrons were initially assigned, we will obtain some FLMOs resembling the bonding MOs. We provide a practical example for an ethane moiety described at the HF/aug-cc-pVDZ level (see Fig. S3). Once the bonding FLMOs are identified (such a step is very easy, since they are the highest in energy among the FLMOs of the considered fragment), the fragment densities of the two fragments covalently bonded can be updated by exploiting Eq. 5 in the main text (i.e. one electron is removed to one fragment density, and added to the other). Therefore, the procedure is independent of the initial electron assignment choice.

S1.3 N-fragment case
In general, at each cycle we are working in a reduced MO space, in which the occupied MOs of the previously optimized density are removed. Eq. S4 can be reformulated in case of N S4 densities, and for which we want to optimized the i-th density: from which the Fock matrix can be constructed: To summarize, the procedure is characterized by the following points: 1. Construction of the G(D) and all the terms Tr h j···N D = Tr (T + N k≥j V k )D. In this way, we have at once all the constant terms due to the total density D 2. Construction of one electron matrices at the i-th step.
3. From the previous cycle Fock matrix construct all the constant terms due to the i−1-th optimized density matrix.
5. Readjust the Fragment Density according to Eq. 5 in the main text.

S1.4 Interaction Energy Decomposition among the FLMOs
To generalize the energy decomposition analysis among the Fragment-Localized Molecular Orbitals (FLMOs), we consider two orbitals α and β localized in fragment A and fragment S5 B, respectively. We assume that the two orbitals are doubly occupied; thus, two density matrices can be computed as: By following the derivation commented in the main text, the interaction between the two densities can therefore be written as: where, the factors n A o and n B o corresponds to the number of occupied MOs of fragments A and B, respectively. Such factors allows for the correct limit when all α and β occupied MOs of the two fragments are summed.

S1.5 Interaction Energies
Dimer The interaction energy for a generic dimer AB, composed of two monomers A and B, is defined as: where E AB is the dimer energy, whereas E vac A and E vac B are the monomers energy in gas-phase (which can in principle be computed by including counterpoise correction). Note that, for S6 covalently bonded fragments, E vac X is calculated at the Unrestricted HF level. Equivalently, Eq. S13 can be recasted in terms of FLMO related quantities (see Eqs. 3,7 in the main text) as: where E AB has been rewritten in terms of A, B energies for the monomers embedded in the dimer (E emb A +E emb B ), and their interaction energy calculated as reported in Eq. 6 in the main text. In Eq. S16, the electronic-preparation energies of the two monomers are introduced.
Such an energetic term takes into account the cost to bring the two monomers from the gas-phase to the final electronic state of the dimer.
Trimer The interaction energy for a generic trimer ABC, composed of three monomers A, B and C, is defined as: By using the same procedure exposed above (which is also valid for a general N fragment case), we obtain: We can now introduce the three-body energy term (∆E 3B ), which reads: Therefore, Eq. S17 can be rewritten as: By then considering that E int,F LM O ABC is additive in the interaction energies between the dimers (see main text), we can compare Eqs. S18 and S20, which yields: and therefore, an expression for the three-body energy term (∆E 3B ) can be obtained: We now move to calculate the interaction energy between two subsystems X and Y, which is defined as: Equivalently, it can be computed by using FLMO-based quantities as: is the electronic preparation energy of the dimer. By using S25, the interaction energy between two dimers can be decomposed in terms of FLMO-based quantities.
It is worth noting that in case the three fragments are covalently bonded, e.g. the hexane molecule depicted in Fig. S2, the gas-phase energies of the different fragments are computed at the UHF level, by specifying the correct multiplicity. On the other hand, E emb X are calculated by following the procedure described in Sec. S1.2 and by exploiting Eq. 5 in the main text.  Table S3: Interaction energy decomposition analysis (kcal/mol) between the three peptides constituting the trimer assembly, prototype of amyloid peptide (see fig. 4 a,b,c).